Test convergence of the series with an = 1/n if n odd and an = 1/n2 if n even

Proof. (This proof might be overkill. You could simply observe that this series is greater than the sum of the reciprocals of the odd integers, and use limit comparison with .)Using the piecewise definition of we have

Then using the limit comparison we have,

Since we know diverges, the limit comparison test establishes the divergence of